# Circle A has a radius of 4  and a center of (5 ,3 ). Circle B has a radius of 5  and a center of (1 ,4 ). If circle B is translated by <2 ,-1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Mar 7, 2017

circles overlap.

#### Explanation:

What we have to do here is $\textcolor{b l u e}{\text{compare}}$ the distance (d) between the centres of the circles to the $\textcolor{b l u e}{\text{sum of radii}}$

• If sum of radii > d, then circles overlap

• If sum of radii < d, then no overlap

Before calculating d we require to find the 'new' centre of B under the given translation which does not change the shape of the circle only it's position.

• " Under a translation "((2),(-1))

$\left(1 , 4\right) \to \left(1 + 2 , 4 - 1\right) \to \left(3 , 3\right) \leftarrow \text{ new centre of B}$

To calculate d, use the $\textcolor{b l u e}{\text{distance formula}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right) \text{ are 2 coordinate points}$

The 2 points here are (5 ,3) and (3 ,3)

let $\left({x}_{1} , {y}_{1}\right) = \left(5 , 3\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(3 , 3\right)$

$d = \sqrt{{\left(3 - 5\right)}^{2} + {\left(3 - 3\right)}^{2}} = \sqrt{4} = 2$

sum of radii = radius of A + radius of B = 4 + 5 = 9

Since sum of radii > d, then circles overlap
graph{((x-5)^2+(y-3)^2-16)((x-3)^2+(y-3)^2-25)=0 [-20, 20, -10, 10]}